A308759 Sum of the second largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 1, 3, 5, 10, 13, 23, 30, 46, 59, 83, 103, 141, 170, 220, 265, 334, 392, 484, 563, 680, 784, 930, 1061, 1247, 1409, 1631, 1836, 2106, 2349, 2673, 2967, 3348, 3699, 4143, 4554, 5077, 5554, 6150, 6710, 7396, 8032, 8816, 9546, 10432, 11264, 12260

Offset

0,7

Links

Table of n, a(n) for n=0..50.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} i.

a(n) = A308775(n) - A308733(n) - A308758(n) - A308760(n).

Conjectures from Colin Barker, Jun 23 2019: (Start)

G.f.: x^4*(1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).

a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.

(End)

Example

Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     10          13          23          30          46        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 07 2019

Mathematica

Table[Total[IntegerPartitions[n, {4}][[All, 2]]], {n, 0, 50}] (* Harvey P. Dale, Nov 08 2020 *)

Crossrefs

Cf. A026810, A308733, A308758, A308760, A308775.

Sequence in context: A345890 A265282 A160792 * A137395 A001767 A360956

Adjacent sequences: A308756 A308757 A308758 * A308760 A308761 A308762

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 22 2019

Status

approved